Posts Tagged ‘Localization’

Thomas Aquinas and other ludicrous pseudo-philosophers (in contradistinction with real philosophers such as Abelard) used to ponder questions about angels, such as whether they can interpenetrate (as bosons do).

Are today’s mathematicians just as ridiculous? The assumption of infinity has been “proven” by the simplest reasoning ever: if n is the largest number, clearly, (n+1) is larger. I have long disagreed with that hare-brained sort of certainty, and it’s not a matter of shooting the breeze. (My point of view has been spreading in recent years!) Just saying something exists, does not make it so (or then one would believe Hitler and Brexiters). If I say:”I am emperor of the galaxy known as the Milky Way!” that has a nice ring to it, but it does not make it so (too bad, that would be fun).

Given n symbols, each labelled by something, can one always find a new something to label (n+1) with? I say: no. Why? Because reality prevents it. Somebody (see below) objected that I confused “map” and “territory”. But I am a differential geometer, and the essential idea there, from the genius B. Riemann, is that maps allow to define “territory”:

Fundamental Idea Of Riemann: the Maps At the Bottom Are Differentiable

Newton later (supposedly) established the equivalence between the 1/dd law and Kepler’s three laws of orbital motion, thus demonstrating the former (there is some controversy as whether Newton fully demonstrated that he could assume planets were point-masses, what’s now known as Gauss’ law).

I insist upon the 1/dd law, because we have no better (roll over Einstein…), on a small-scale.

Laplace (and some British thinker) pointed out in the late 18C that this 1/dd law implied Black Holes.

The third ingredient in the annihilation of infinity was De Broglie’s assertion that to every particle a wave should be associated. The simple fact that, in some sense a particle was a wave (or “wave-packet”), made the particle delocalized, thus attached to a neighborhood, not a point. At this point, points exited reality.

Moreover, the frequency of the wave is given by its momentum-energy, said De Broglie (and that was promptly demonstrated in various ways). That latter fact prevents to make a particle too much into a point. Because, to have short wave, it needs a high frequency, thus a high energy, and if that’s high enough, it becomes a Black Hole, and, even worse a Whole Hole (gravity falls out of sight, physics implodes).

@Patrice Ayme: It seems that you are making the exact same conflation of “the map” and “the territory” that I’ve recommended should be avoided. There is no such thing as the largest number in our conceptual model of numbers, but there is at any given point, a limit on the number of particles in the physical universe. If tomorrow we find that each fermion consists of a million vibrating strings, we can easily accommodate the new limit because of the flexible conceptual structure provided by the infinite assumption in our mathematics.

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I know very well the difference between “maps” and territory: all of post-Riemann mathematics rests on it: abstract manifolds (the “territories”) are defined by “maps Fi” (such that, Fi composed with Fj is itself a differential map from an open set in Rx…xR to another, the number of Real lines R being the dimension… Instead of arrogantly pointing out that I have all the angles covered, I replied:

Dear Pradeep Mutalik:

Thanks for the answer. What limits the number of particles in a (small enough) neighborhood is density: if mass-energy density gets too high, according to (generally admitted) gravity theory, not even a graviton could come out (that’s even worse than having a Black Hole!)

According to Quantum Theory, to each particle is associated a wave, itself computed from, and expressing, the momentum-energy of said particle.

Each neighborhood could be of (barely more than) Planck radius. Tessellate the entire visible universe this way. If too each distinct wave one attaches an integer, it is clear that one will run out of waves, at some point, to label integers with. My view does not depend upon strings, super or not: I just incorporated the simplest model of strings.

Another mathematician just told me: ‘Ah, but the idea of infinity is like that of God’. Well, right. Precisely the point. Mathematics, ultimately, is abstract physics. We don’t need god in physics, as Laplace pointed out to Napoleon (“Sire, je n’ai pas besoin de cette hypothese”). (I know well that Plato and his elite, tyrant friendly friends and students replied to all of this, that they were not of this world, a view known as “Platonism”, generally embraced by mathematicians, especially if they are from plutocratic Harvard University… And I also know why this sort of self-serving, ludicrous opinion, similar to those of so-called “Saint” Thomas, a friend of the Inquisition, and various variants of Satanism, have been widely advocated for those who call for self-respect for their class of haughty persons…)

The presence of God, aka infinity, in mathematics, is not innocuous. Many mathematical brain teasers become easier, or solvable if one assumes only a largest number (this is also how computers compute, nota bene). Assuming infinity, aka God, has diverted mathematical innovation away from the real world (say fluid flow, plasma physics, nonlinear PDEs, nonlinear waves, etc.) and into questions akin to assuming that an infinity of angels can hold on a pinhead. Well, sorry, but modern physics has an answer: only a finite number.

Human brains are built from ideas. Any change in such ideas is lots of work, thus pain, and is always resisted. Often viciously. The greater the change, the more vicious the backlash.

A contributor, “Disagreeable Me” (who had published an extensive essay on consciousness, Sept 1, 2014) rose strident objections to my thesis (found in preceding comments; such stridency is not new: I am used to violent critiques against Quantum Consciousness, in the last few decades that I have dragged this pet around). Here is some of the dialogue, raw (co-sent to Scientia Salon):

Disagreeable Me: “Most people seem to assume that their consciousness is in some way located in their brains. Personally, I agree with you that it is not a localized thing, but this is because I think consciousness is a property of a mind, and that a mind is an abstract object.

That’s quite different meaning of the word, however. In quantum mechanics, non-locality means that effects seem to work instantaneously at a distance. I don’t see any reason for believing that consciousness has these attributes unless you want to bring up woo such as remote viewing or clairvoyance or mind-reading.”

Patrice: One could argue that all “objects” are “abstract” (or at least abstractions, in the mathematical sense Alonso Church gave that in the 1930s; Church was Turing’s thesis adviser). Abstraction is characterized by the stripping of secondary, inessential characteristics. So one may, indeed, loose localization. That’s vague (joke intended: vague = wave -> delocalized).

However, my point about localization is different. And precise. Brain delocalization is biologically grounded. The brain is, physiologically, a delocalized object.

The brain is made of many neighborhoods, and subsystems. Is the brain the temporal lobe? The cerebellum? The right brain? The frontal cortex? Clearly much of the brain is working all over, much of the time. Some parts get active, others go to sleep, other parts never stop (say those watching over basic functions such as breathing or neurohormonal cycles).

So, when we consider the brain, we consider something spatially spread out. Yet, the conscious feeling that emanates from it, what we call consciousness, somehow, is centralized. Consciousness is one, not multiple, not spread out, at any instant of time.

How to make one, out of many? This is a question that arises naturally when considering both brain, and consciousness.

One could object that the same can be said about a bridge. A bridge is an abstraction of many characteristics. Yet, what makes the perception of a bridge one? Consciousness.

If one focuses on one’s breathing and heart rate, as conscience can do, and commands them, the mind is then just about that. Conscience focuses on a (few) characteristic(s). One could say that conscience collapses on particular points.

Now think about the way a Quantum process enfolds: it’s about something wavy spread about that is processed, to become, in the end, just one.

This sole sentence abstracts the basic set-up of Quantum physics: “something wavy”: the wavefunction, the “spread about” is a Hilbert space; “processed” is about time as an evolution parameter; “in the end” is about collapse/decoherence; “the one” is the so called “particle state” that results.

The analogy with the contrast of the delocalized brain in an union with a focused, localized consciousness, free to localize inside the brain wherever it wills, jumps at me.

DM: …”the following sentence makes your meaning clearer. “If consciousness were not Quantum, it would have to be “classical”, that is, not fundamental.” So, you’re argument is that everything that is fundamental is quantum, and it is completely stupid to imagine that consciousness is not fundamental.

This is largely meaningless to me. I don’t know what you mean by fundamental, and it is not obvious to me that everything that is fundamental is quantum. I might, for instance, claim that logic (i.e. the law of non-contradiction) is fundamental, but it would seem to be very strange to claim that logic is Quantum, whatever that would mean.”

Patrice: That’s indeed my argument. Although it’s not yet clear how exactly, all of Classical Mechanics, Relativity, and Thermodynamics have to emerge from Quantum Physics, I believe. I would call that Ultimate Unification (UU). (GUT, Grand Unified Theories, are less ambitious: they unify only at high energies; UU is a conjecture, right, but so is Langlands program in mathematics; nobody sneers at that.)

Right now, experimental research is exploring the transition from QM to CM, and has been honored with the 2012 Nobel Prize. (Haroche in Paris, for counting photons without disturbing them, and his colleague Wineland in Boulder, for doing quantum computing with ions, among other things.) We are very far from a full picture on how to implement UU (the Nobel committee recognized Haroche and Wineland’s works as first timid steps to the Quantum computer).

Logic is a vast subject. In 1936, two of the most advanced mathematicians (Birkhoff and Von Neumann) invented something they called Quantum Logic, doing away with the distributive law. I do not doubt, though, that logic is a form of empiricism (whether the one gets from reality, or… the imagination).

It’s curious that you mention the law of non-contradiction as fundamental (as Aristotle held, in contradiction with Heraclitus). Quantum Physics is well known to enjoy things that are alive and dead simultaneously. It seems rather contradictory to me that some don’t appreciate the contradiction.

DM: “What you call freedom I call randomness. Randomness is not freedom, but if nature is indeterministic then all objects are random anyway. Chaos theory suggests that small perturbations in complex systems such as brains can lead to radically different outcomes. “

Patrice: Agreed. Except that I do not confuse freedom and randomness. Randomness can help freedom, and vice versa, but they are not the same. Schopenhauer famously claimed he could not will what he willed. I beg to disagree: the wise will will what she wills, such is her definition. Higher reflectivity, and detachment from contingency, is what intelligence is all about.

I thank Disagreeable Me for giving me the occasion to become more conscious in the matter of consciousness (and offering me the occasion to make a quantum jump of understanding, etc.)